Poset representations of distributive semilattices

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We prove that for any distributive join-semilattice S, there are a meet-semilattice P with zero and a map f:PxP-->S such that f(x,z)<=f(x,y)vf(y,z) and x<=y implies that f(x,y)=0, for all x,y,z in P, together with the following conditions: (i) f(y,x)=0 implies that x=y, for all x<=y in P. (ii) For all x\leq y in P and all a,b in S, if f(y,x)=avb, then there are a positive integer n and a decomposition x=x_0<=x_1<=...<=x_n=y such that f(x_{i+1},x_i) lies either below a or below b, for all i < n. (iii) The subset {f(x,0)|x\in P} generates the semilattice S. Furthermore, any finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive (v,0,1)-semilattices.

Mots clés: Poset; distributive; semilattice; p-measure; diagram; relatively complete extension; interval extension; covering extension; normal interval diagram; standard interval scheme; strong amalgam; doubling extension

Domaines: Anneaux et algèbres [math.RA]; Mathématiques générales [math.GM]

Friedrich Wehrung. Poset representations of distributive semilattices. 2006. ⟨hal-00016421v2⟩